## 1. Introduction

This paper addresses the growth and decay of a convective boundary layer (CBL) over a surface with a constant surface temperature. Few studies exist of the basic properties of such a boundary layer; most of the simulation-based studies of the properties of the turbulent flow in the CBL have been done using a fixed surface flux (e.g., Moeng 1984; Sullivan et al. 1998; Fedorovich 1995; Garcia and Mellado 2014), whereas many of the studies with a fixed surface temperature address cloudy boundary layers over sea, often in a setting including radiative cooling (e.g., Tompkins and Craig 1998; vanZanten et al. 2011).

Our study of a CBL over a fixed-temperature surface has relevant applications. It represents, for instance, a limiting case of the decay of turbulence in the CBL: one where the system dies out very slowly. Furthermore, it can help in understanding heterogeneously heated and cooled boundary layers over sea ice; here, it represents the limiting case of a boundary layer that forms over a very wide Arctic lead (Esau 2007). It also represents an idealized setting to study the reaction of a CBL over sea to changes in the sea surface temperature.

In this paper, we study this CBL in one of its most simple forms: the growth of a CBL against a linear stratification over a surface with a fixed temperature, without a balancing cooling force and without a large-scale horizontal pressure force and subsidence. This system develops and dies out over time, as the atmospheric buoyancy evolves toward that of the surface, resulting in an unsteady system with a vanishing near-surface gradient. Our aim is to derive a mathematical model for the system in order to find the relevant time scales as a function of the external parameters. This model is verified against direct numerical simulations (DNSs) of the system. The motivation for using DNS is that it does not require us to use a surface model, as its applicability in a turbulence resolving model under conditions of free convection is still under discussion (Zilitinkevich et al. 2006; Mellado et al. 2015). As the Reynolds numbers acquired in direct numerical simulations are several orders of magnitude smaller than those in the atmosphere, this study contains a careful assessment of whether the results can be extrapolated to the atmosphere.

The organization of this paper is the following: In section 2, we define the system and apply dimensional analysis to minimize the number of independent parameters and to obtain first estimates of characteristic scales. Subsequently, we describe the simulation setup in section 3 and study the evolution of the system based on the simulations in section 4. In section 5, we propose a mathematical model that describes the evolution of the system as a function of time and derive the analytical solutions, which we verify against the simulations in section 6. In this analysis, we explore the presence of self-similarity and Reynolds number similarity. This is followed by a discussion on the applicability of the results to the atmosphere in section 7, including a discussion on the importance of surface roughness and of the relevance of this case to the study of the decay of turbulence during the afternoon transition.

## 2. Physical model and dimensional analysis

The physical model being studied is a linearly stratified atmosphere with kinematic viscosity *Î½* and thermal diffusivity *Îº* that is heated from below by a surface with a constant temperature (Fig. 1). For generality, we make use of buoyancy as our thermodynamic variable. We define buoyancy in terms of virtual potential temperature as ^{âˆ’2}. We express the initial linear stratification as parameter

*Îº*and

*L*:which is the height at which the buoyancy of the linearly stratified atmosphere equals the surface buoyancy (Fig. 1). This is the maximum size the system can achieve. Then, we define the surface buoyancy flux scale

*B*:following inner-layer scaling (Townsend 1959; Mellado 2012), where we assume that the influence of stratification is not felt in the surface layer. With the help of the previous two scales, we can define a velocity scale to normalize the velocity fluctuations caused by turbulence:following Deardorff (1970). With the defined length and velocity scales, one can rewrite the defined Reynolds number according to the classical definition

*Î·*for this system, where we assume that the total dissipation in the system is a fraction, of order 1, of the surface buoyancy flux. This giveswhere the unity value of the Prandtl number is used to substitute

*Î½*with

*Îº*. It follows that the Reynolds number is equal to the four-thirds power of the scale separation; thus,

*T*to be proportional to the time it takes to warm up the system from the initial linear stratification to buoyancy

*L*. The total energy required is proportional to

*T*. The total added energy is the equal to the integral in Eq. (7). If this integral is assumed to be a right triangle with legs

## 3. Numerical simulations

### a. Formulation and model description

*b*, and volume, formulated in flux form for a Boussinesq fluid:where

*Ï€*is a modified pressure.

The velocity boundary conditions are specified as no penetration *L*.

We use MicroHH (http://microhh.org), which is a 2D-parallel combined DNS/LES code. Fully conservative, fourth-order-accurate finite-difference schemes (Morinishi et al. 1998; Vasilyev 2000) have been used, combined with a low-storage third-order Rungeâ€“Kutta time integration scheme (Williamson 1980). The pressure is acquired by solving a Poisson equation. Here, the horizontal dimensions are decoupled using a Fourier decomposition, and for each mode a heptadiagonal matrix is solved. In the top of the domain (upper 25%) a damping layer is applied that prevents the reflection of gravity waves back into the domain with a damping time scale that is infinity at the bottom boundary of the damping layer and decreases exponentially to

### b. Numerical experiments

The results in this study are based on four direct numerical simulations, with varying Reynolds number and an identical Prandtl number of unity (Table 1). Each simulation has been run at a horizontal domain size of 2 m, with a linear stratification ^{âˆ’2}. The variations in the Reynolds number are acquired by varying the surface buoyancy *Îº*. As the acquired boundary layer height is well approximated by

Overview of the numerical simulations.

## 4. Results

### a. Characteristics

*Ï•*is an arbitrary variable. The value of

*b*âˆ’

*N*

^{2}

*z*, of kinetic energy

*Îµ*, denoted as

The time evolution of the mean surface buoyancy flux

The time evolution of the boundary layer depth *L*. The time evolution of those simulations is well predicted by the model.

The time evolution of the vertically integrated mean buoyancy (Fig. 2c) shows two lines for each simulation: namely, the surface contribution to the integral, calculated as

The time evolutions of vertically integrated mean kinetic energy, buoyancy flux, and dissipation (Figs. 2dâ€“f) demonstrate the complexity of the system. The time evolution of the integrated kinetic energy (Fig. 2d) shows that there is first a phase in which the kinetic energy increases, during which the added potential energy through the surface buoyancy flux is converted into kinetic energy and the benefits of a deeper CBL outweigh the loss of buoyancy supply at the bottom boundary. The buoyancy flux and dissipation show a similar pattern, but with a peak that occurs earlier in time.

After the peak, the integrated variables decrease in time and slowly develop toward zero. Similar to the time evolution of the surface buoyancy flux, also the time evolution of the three kinetic-energy-related variables shows a complex decay pattern that is neither exponential nor follows a power law. With the model that we derive in section 5, we provide the proper algebraic scaling.

The derived model only predicts the correct kinetic energy for the two cases with the highest Reynolds numbers. Interestingly, the buoyancy flux and the dissipation are adequately predicted by the derived model in all four simulations. This suggests that, in the simulations with low Reynolds numbers (ReS and ReM), there is insufficient scale separation between the large scales at which the production happens and the smaller scales at which the energy is dissipated. Plumes are therefore already dissipated before they can reach their full potential, resulting in a lower integrated kinetic energy than in simulations ReL and ReXL.

We conclude from the analysis that a high Reynolds number is a requirement for the study of the initial peak in integrated kinetic energy and, even more so, of those in the integrated buoyancy flux and dissipation. Only then is the time it takes to forget the initial perturbations and to form a fully turbulent layer sufficiently shorter than the time it takes to form the peak in kinetic energy so that the model is able to predict the integrated kinetic energy during the phase that its magnitude is still steeply increasing. The evolution toward the peak is thus not related to the spinup of the model, but is a fully physical process.

### b. Reynolds number

_{*}), and the second is the maximum in the vertical profile of the Taylor Reynolds number (Re

_{Î»}), as defined by Pope (2000, p. 200). In the latter, we substituted

*Î»*with

_{*}is the height of the minimum in the mean buoyancy flux

_{Î»}, ReL has its peak at a value around 150 and decreases subsequently to values close to 100, whereas ReXL peaks close to 180. Both simulations fulfill the criteria of Dimotakis (2000) for fully developed turbulence, which sets an Re

_{Î»}of 100â€“140 as the threshold; thus, this corroborates our observation of Reynolds number similarity for simulations ReL and ReXL (Fig. 2).

To illustrate the influence of the Reynolds number on the flow characteristics, two cross sections of the surface buoyancy flux are displayed in Fig. 4. The left cross section shows simulation ReL at the moment of maximum Reynolds number, whereas the right one shows the state of that variable at the end of the simulation, where the Reynolds number has decreased considerably. The decrease in Reynolds number reveals itself in the loss of small-scale features in the flow, as the size of the smallest eddies that can exist is increasing because of the decrease of energy input at the largest scales.

## 5. Mathematical model

### a. Governing equations of the model

### b. Solutions of the governing equations

*t*to

### c. Fitting the constants

## 6. Scaling the results

### a. Time evolution

The scaled time evolutions of the buoyancy- and kinetic-energy-related variables are shown in Fig. 6, which is similar to Fig. 2, but with the mathematical model used to normalize the results. For each of the variables, a constant value in time corresponds to a perfect performance of the model. Each of the results show a convergence toward the value predicted by the model with increasing Reynolds numbers, and for all plotted variables there is convergence for simulations ReL and ReXL.

We can conclude from the time evolution of scaled surface buoyancy flux

The height evolution (Fig. 6b) shows a scaled height

The scaling of the kinetic-energy-related variables is more subtle. The model is able to predict the kinetic energy well (Fig. 6d), but the kinetic energy is sensitive to the correct prediction of the surface buoyancy flux; it drops quickly as soon as

### b. Profiles

In Fig. 7, we present scaled vertical profiles of relevant buoyancy- and kinetic-energy-related variables, making use of the scaling variables provided by the derived model. The profiles are taken at equal intervals in the range where the nondimensional time of the system

Similar conclusions can be drawn from the profiles of the mean buoyancy flux

The vertical profiles of the kinetic energy and the velocity variances show that the Reynolds number of the flow has a much larger impact on the kinetic-energy-related flow properties than on the thermodynamic characteristics of the flow. In Fig. 2, which displayed the time evolution of the vertically integrated mean kinetic energy, we found that the total normalized kinetic energy roughly doubles from the lowest to the highest Reynolds number. Figures 7dâ€“f shows that the increase in kinetic energy with larger Reynolds numbers comes with a change in the shape of the profiles, where the two largest Reynolds numbers recover the profiles with the shape and magnitudes, exactly as those in Sullivan and Patton (2011, their Fig. 6). Note the conversion factor

To conclude, the analysis of the vertical profiles of buoyancy and velocity variances validates the applicability of our mathematical model for the scaling of highâ€“Reynolds number simulations. Furthermore, it shows that, even though the integrated kinetic energy exhibits a complex time evolution, the vertical profiles display self-similarity. The Reynolds number similarity displayed in the two cases with the highest Reynolds numbers encourages the use of direct numerical simulation as a tool in atmospheric turbulence, as the required resolution to recover converged results from large-eddy simulations is only marginally higher [Sullivan and Patton (2011) found similarly converged results at

## 7. Discussion: Roughness and decay during the afternoon transition

In the previous sections, we have validated the derived characteristic scales and the mathematical model for the bulk characteristics of the CBL and several kinetic-energy-related variables and have shown the presence of Reynolds number similarity. This allows us to extrapolate the results to the atmospheric boundary layer with very high Reynolds number. In this section, we use the mathematical model to analyze the time evolution of the system under typical atmospheric conditions. We have chosen here for an excess temperature of 6 K, a lapse rate of 0.006 K m^{âˆ’1}, a thermal diffusivity of 1 Ã— 10^{âˆ’5} m^{2} s^{âˆ’1}, and a buoyancy parameter ^{âˆ’2} K^{âˆ’1}.

One important difference between our experiments and most atmospheric flows is the type of bottom boundary, as nearly all atmospheric flows are rough. Zilitinkevich et al. (2006) and Beljaars (1995) have shown that free convection over rough surfaces is a delicate issue and that full understanding is still lacking. Nonetheless, Zilitinkevich et al. (2006) has estimated that, over a rough surface, the transfer coefficient can increase two orders of magnitude compared to a smooth surface. To study the time scales in an approximate atmospheric setting, we have calculated the time evolution of the system using the fitted constant ^{âˆ’2}. All three are shown in Fig. 8.

According to the scaling laws, the rate of change of the system is proportional to

The linear dependence of the time scales in our system on the buoyancy transfer rate makes the surface model a crucial, and potentially overlooked, model component under conditions of free convection. The exact behavior of free convection over a rough surface is still not fully understood, although adequate parameterizations for large-scale models have been developed (Beljaars 1995). These solutions, however, are not applicable in large-eddy simulations, where the large-scale motions of the size of the CBL depth are resolved. Zilitinkevich et al. (2006) have made the case that, in regions of horizontal flow toward plumes, Moninâ€“Obukhov-like parameterizations are applicable. However, few measurement data are available, and the solution to this problem remains incomplete. Consequently, large-eddy simulations of (cloudy) boundary layers over sea surfaces may exhibit an important dependence on the chosen formulation of the surface model and the way roughness is accounted for during free convection.

With respect to the decay of turbulence during the afternoon transition, our results indicate that, at atmospheric Reynolds numbers, the quasi-steady state, thus the dominant balance between the buoyancy flux and dissipation, can be maintained until the input of energy from the surface buoyancy flux has nearly vanished. Van Driel and Jonker (2011) have shown that this balance holds for systems that have slow fluctuations in the surface buoyancy flux, although they worked with prescribed fluxes. As a result of this balance, the time evolution of the kinetic energy in the system can be excellently predicted from the solution of Eq. (10), as long as the appropriate model for

## 8. Conclusions

We have characterized the growth and decay of a convective boundary layer (CBL) over a surface with a constant surface temperature and a linear stratification. This system has only the Reynolds and Prandtl numbers as nondimensional parameters. We have done direct numerical simulations for four different Reynolds numbers and have chosen a Prandtl number of unity for all simulations. We have derived a mathematical model that describes the time evolution of the buoyancy- and velocity-variance-related variables and verified the model against the simulations.

Each simulation has a decaying surface buoyancy flux from the beginning, because the temperature difference between the surface and the atmosphere is decreasing. However, the vertically integrated kinetic energy, buoyancy flux, and dissipation initially increase in time, because the contribution of boundary layer growth is more important than the decay of the flux. These variables develop toward a peak and decay subsequently. The derived model is very well able to describe the evolution of the bulk variables of highâ€“Reynolds number flows. Our simulations display Reynolds number similarity for the two cases with the highest Reynolds numbers, which suggests that our results can be extrapolated to the atmosphere, despite their moderate Reynolds numbers. This demonstrates the applicability of direct numerical simulation to the study of atmospheric boundary layers.

The time rate of change of the system is linearly related to the surface flux of buoyancy, and therefore any atmospheric model study depends crucially on correctness of the mathematical formulation of the surface model. Especially in large-eddy simulations over water surfaces, which is a common setting for studies of cloudy boundary layers, the importance of the chosen surface roughness may have been underestimated. Even in the case of a friction-velocity-dependent roughness (Charnock 1955), an arbitrary constant is involved that has a large influence on the time scale of the system. We, however, cannot give the definitive answer on the role of surface roughness in free convection, which warrants a repetition of this study, but with a rough surface implemented at the bottom boundary.

This system can be seen as a limiting case for the decay of turbulence during the afternoon transition, as the surface flux slowly develops toward a value of zero. Our results show that the evolution of kinetic energy in the decay phase is not exponential, nor does it follow a power law, as a result of the competing effects of boundary layer growth and a decreasing surface flux. The derived model in this paper is able to predict the correct evolution in time of the CBL depth, kinetic energy, buoyancy production, and dissipation.

## Acknowledgments

Support from the Max Planck Society through its Max Planck Research Groups program is acknowledged. Computational resources were provided by the JÃ¼lich Supercomputing Centre. We acknowledge Alberto De Lozar for the discussions on the scaling of the data and Antoon van Hooft for his useful comments on the paper.

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